Jon Deeble, the Red Sox first base coach and recruiting agent, said the team had seen video of Gilchrist's impressive abilities as a batsman and offered to hold a personal assessment session with him to see if the skills could be transferred to a baseball diamond.
"They have expressed interest, there is no doubt about that," Deeble told Channel Seven television on Thursday.
The left-handed wicketkeeper has a reputation as one of the cleanest hitters in cricket history, but at the age of 33 he seemed an unlikely target for US baseball's reigning champions.
"They have seen videotapes of him hitting a cricket ball, he's got the two key ingredients, he has got the eye and the power," said Deeble, who also coaches Australia's Olympic baseball team.
Gilchrist's manager, Stephen Atkinson, told The Daily Telegraph newspaper that the player had been flattered by the Red Sox approach but was not tempted to leave the cricket pitch, just yet at least.
"Cricket remains (Gilchrist's) number one priority and the approach may not come to anything, but it is genuine," he said.
"If we did anything, it would have to fit in with his cricket commitments," he said.
Atkinson dismissed suggestions the Red Sox offer was not serious, especially given Gilchrist's age.
"It is not pub talk or someone day dreaming," he said. "It is (Deeble's) job to identify players and he likes the look of Adam."
Major league baseball salaries dwarf those earned by cricketers, even on Australia's World Number one team. Top members of Australia's cricket team earn about one million dollars ($760,000) per year in salary, where the average US baseball player's pay is more than $2.5 million.